Weak Stability of Multidimensional Shocks

Jean-François Coulombel ()
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Jean-François Coulombel: ENS Lyon

A chapter in Hyperbolic Problems: Theory, Numerics, Applications, 2003, pp 473-481 from Springer

Abstract: Abstract In [8], A. Majda studied the linear stability of multidimensional shock waves for general systems of conservation laws. His analysis relied on two main assumptions: the shock wave was assumed to satisfy the so-called uniform stability condition (we shall recall it in the sequel of this paper) and the system was assumed to satisfy a block structure condition. The linear study performed in 8 enabled Majda to prove the (local) existence of multidimensional shock waves in this context, see [7, 9, 14] for an overview. Up to our knowledge, the study of multidimensional shock waves has known little progress since these oustanding breakthroughs. Let us mention however two significant results. To carry out the study of the variable coefficients linear systems in [8], Majda used an H 8 version of pseudodifferential calculus (essentially based on Moser’s inequalities). Another approach for this type of problems is the paradifferential calculus of Bony and Meyer [2, 12]. Using these techniques, Mokrane precised the regularity assumptions under which Majda’s theorems still hold, see [13] and [11]. More recently, Métivier has shown in [10] that the block structure condition defined by Majda in [8] was met by all symmetric hyperbolic systems with constant multiplicity. Whether the block structure condition is met by the magnetohydrodynamics equations (or by other nonconstant multiplicity systems) is still an open (but interesting) question.

Keywords: Shock Wave; Unstable Mode; Weak Stability; Planar Shock Wave; Magnetohydrodynamics Equation (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55711-8_43

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DOI: 10.1007/978-3-642-55711-8_43

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